How do I initiate contact with academics working on elliptic curves despite being very new to the subject. Could I ask them to explain something that would be found in a textbook?
Unlike the other individual points of view expressed here, I think this can certainly be done in a way that neither rude, spam, nor unwise.
While there are certainly "I can't be bothered" types in any academic field, there are also plenty of folks who understand that the finding various ways to explain their research, especially to people with different levels of familiarity is a valuable exercise for scholars and some would even see it as a sacred duty1.
Enlightened scholars will also know that sometimes unexpected benefits arise from unexpected or nonstandard discussions.
I would not say that if they refuses or ignored a well-written email of this type that they were "bad" or unenlightened, but I would see their willingness to discuss with you or not as a sort of filter for the kind of person you'd like to get to know in the first place!
Okay, but what would a "well-written email of this type" be like?
It should be honest and upfront but not off-putting. Explain that you don't want a full lecture on the topic, but before you take the dive you want a little bit of guidance and/or clarification to make sure you get started on the right foot and path. Include a few specifics that make it clear you have at least some understanding of the topic and that they would not be wasting their time.
Mention a time limit, like 15 or 20 minutes. Of course if they feel it's productive they may extend, but make it clear you have every intention of keeping it short and promptly leaving if they don't offer to extend. You want to give them an opportunity to see if they can help, without fear that they have a lot to loose.
- If you get a positive response2, then despite what you said, study like crazy and make at least some progress so that they might be pleasantly surprised that you were a bit modest describing your level of understanding, and so that they are further reassured that spending a little time with you has a chance of making a real difference. To that end cgb5436's answer suggests some resources to start with.
- If you get a negative response send a short, concise thank you reply and follow up on your own on any advice they might have included in their response (unless it's bad advice!)
- If you get no response then you should not pursue it further with that person at this time. While occasionally mail messages get missed or move to the spam folder, most likely they don't want to deal with the situation, and you don't want to do anything that looks troublesome or harassing or else word of you may spread in a negative way.
- 1in moderation.
- 2and even if you don't!